Question

If the periodic function f(x) satisfies the equation f(x + 1) + f(x –1) = √3 f(x) ∀ x R ∈ then find the period of f(x).

Answer 1

Replacing x by x + 2 in equation (4) then f (x + 4) + f (x) = f (x + 2) ...(5)

Adding equations (4) and (5), we get f(x + 4) + f (x – 2) = 0 ...(6) 

Again replacing x by x + 6 in (6) then f (x + 10) + f (x + 4) = 0 ...(7) 

Subtracting (6) from (7), we get f (x + 10) – f (x – 2) = 0 ...(8) 

Replacing x by x + 2 in (8) then f (x + 12) – f(x) = 0 or f (x + 12) = f(x) 

Hence f(x) is periodic function with period 12.